Is interaction possible without heredity?

Abstract

Most particles we know seem to interact through fields. Propagation of fields with a finite velocity leads to hereditary properties. For instance, if two electrons interact by means of retarded potentials it is well known that positions and velocities given at a fixed t ime are not sufficient data for prediction. In order to determine the further motion of the system, we need information about its past behaviour (1). Even several at tempts based upon an action-at-distance point of view exhibit hereditary features (2). However, i t is of interest to ask if nonhereditary relativistic dynamics is at least conceptually possible for interacting points. So far, many no-interaction theorems (3) have discarded any hope of Hamiltonian formulation. (At least, if there were agreement about the meaning of Hamil tonian here !) But one can still investigate from a more general point of view, asking if, besides hereditary systems, dynamics involving a finite number of init ial data are also possible. While hereditary interacting systems more or less satisfy a strong causality principle based upon light cones, we only require that the systems we look for are causa l i n the sense o] the im~tial-data p r o b l e m . This problem must be solvable without preferring any spacelike hypersurface. The above considerations lead to the following axioms (4):